**1 Answer**

written 5.5 years ago by |

**Express (127.125)10 in the IEEE single precision standard of floating point representation:**

**Step 1: Convert the decimal number to its binary fractional form.**

Convert this format is in base2 format For this, first convert 127 in to binary format

127=1111111

Convert 0.125 in to binary format

0.125*2=0.250 0

0.250*2=0.500 0

0.500*2=1.00 1

Binary format of 127.125=1111111.100

Shifting this binary number

1.111111100*2 **6 Normalized

1.111111100 is mantissa

2 **6 is exponent

add exponent 127 +6=133

**Step 2: Normalize the binary fractional number.**

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.111111100 times 2*6

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

**Step 3: Convert the exponent to 8-bit excess-127 notation**

Add 127 to the exponent and convert it to 8-bit binary:

6+ 127 = 133--> 10000101

**Step 4: Convert the mantissa/significand to "hidden bit" format.**

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.111111100--> 111111100

**Step 5: Write down the 1+8+23 = 32 bits.**

127.125 is positive - the sign bit is zero: 0

The next eight bits are the exponent: 10000101

The next 23 bits are the mantissa: 11111110000000000000000

Binary result (32 bits): 01000010111111110000000000000000

**Express (127.125)10 in the IEEE double precision standard of floating point representation.**

**Step 1: Convert the decimal number to its binary fractional form.**

Convert this format is in base2 format For this, first convert 127 in to binary format

127=1111111

Convert 0.125 in to binary format

0.125*2=0.250 0

0.250*2=0.500 0

0.500*2=1.00 1

Binary format of 127.125=1111111.100

Shifting this binary number

1.111111100*2 **6 Normalized

1.111111100 is mantissa

2 **6 is exponent

add exponent 1023+6=1029

**Step 2: Normalize the binary fractional number.**

Move the decimal point left or right so that only a single binary

digit "1" is to the left of the binary decimal point. Compensate by

adjusting the exponent in the opposite direction.

1.111111100 times 2*6

Moving the decimal left seven decreases the size of the number; so,

we use an exponent of 6 to compensate and keep the number the same size.

**Step 3: Convert the exponent to 8-bit excess-127 notation**

Add 1023 to the exponent and convert it to 8-bit binary:

6+1023= 1029--> 10000000101

**Step 4: Convert the mantissa/significand to "hidden bit" format.**

Since every binary floating-point number (except zero!) is normalized

with "1." at the start, there is no need to store that leftmost "1".

Remove the leading "1." from the mantissa/significand:

1.111111100--> 111111100

**Step 5: Write down the 1+11+52 = 64bits.**

127.125 is positive - the sign bit is zero: 0

The next 11 bits are the exponent: 111111100

The next 23 bits are the mantissa: 111111100000000000000000000000000000000000000000000000

Binary result (64 bits): 0111111100 111111100000000000000000000000000000000000000000000000